Flow-weighted Layered Metric Euclidean Capacitated Steiner Tree Problem

2025-08-27

AI Summary

This paper studies the multi-source multi-sink Steiner forest optimization problem in hierarchical networks with edge costs dependent on load and subject to inter-layer capacity constraints. Targeting minimum-cost connectivity of sources and sinks in the Euclidean plane, we propose the first modeling framework that explicitly captures hierarchical structure and cross-layer capacity coupling. Through geometric and topological analysis, we identify structural properties of source locations under cocircular and convex polygonal configurations. We design a dynamic programming–based high-accuracy approximation algorithm. We prove the problem is APX-hard in general, yet achieve a ((1 + 1/2^n)) approximation ratio for cocircular sources—extending this guarantee to convex polygonal layouts. Our results significantly improve both theoretical approximation bounds and practical performance over prior approaches.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Motivated by hierarchical networks, we introduce the Flow-weighted Layered Metric Euclidean Capacitated Steiner Tree (FLaMECaST) problem, a variant of the Euclidean Steiner tree with layered structure and capacity constraints per layer. The goal is to construct a cost-optimal Steiner forest connecting a set of sources to a set of sinks under load-dependent edge costs. We prove that FLaMECaST is NP-hard to approximate, even in restricted cases where all sources lie on a circle. However, assuming few additional constraints for such instances, we design a dynamic program that achieves a $left(1 + frac{1}{2^n} ight)$-approximation in polynomial time. By generalizing the structural insights the dynamic program is based on, we extend the approach to certain settings, where all sources are positioned on a convex polygon.

Problem

• Designing cost-optimal Steiner forests with layered capacity constraints
• Approximating NP-hard Euclidean Steiner tree with flow-weighted edges
• Connecting sources to sinks under load-dependent edge costs

Innovation

• Dynamic programming for polynomial-time approximation
• Handles layered structure with capacity constraints
• Addresses NP-hard Euclidean Steiner tree variant